Evolving Notions of Geometry in String Theory

Abstract

The unfolding of string theory has led to a successive refinement and generalization of our understanding of geometry and topology. A brief overview of these developments is given.

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Notes

  1. 1.

    In other words, the low end of the spectrum of string resonances had a ‘tachyonic’, negative m 2. Here, the term ‘tachyonic’ refers to vacuum instability, not faster-than-light propagation.

  2. 2.

    Up to systematic corrections in powers of the curvature in string units, small when the radius of curvature is large relative to the string scale s .

  3. 3.

    Notation: X μ(ξ), μ=1…d specify the location of the worldsheet point labelled by ξ in the ambient spacetime with coordinates X; dX is the gradient with respect to local coordinates ξ on the worldsheet; G μν and B μν are the tensor components of the metric and antisymmetric tensor fields; Φ is the scalar dilaton; and R(ξ) is the intrinsic local curvature density on the worldsheet.

  4. 4.

    String resonances are gapped with a characteristic level spacing β H ħℓ s /c; the density of states grows as exp[β H E].

  5. 5.

    Momenta on circle are gapped with a spacing ħc/R, while windings on a circle increment in multiples of \(\hbar c R/\ell_{s}^{2}\).

  6. 6.

    In non-supersymmetric examples [29], there are typically ‘tachyonic’ states among the twisted sector strings, meaning the vacuum is unstable to the formation of a condensate of these strings. The decay of the vacuum automatically smooths the singularity.

  7. 7.

    By a process known as a flop transition [32].

  8. 8.

    If the spatial dimension of the brane exceeds the cycle dimension by one, tensionless strings arise. Tensionless membranes and higher dimensional objects seem not to occur [37]; the energetics is such that either light strings or light particles dominate the low-energy dynamics in any such ‘singular’ limit.

  9. 9.

    Indeed, in certain limiting situations, the D-branes are solitons [38].

  10. 10.

    The fact that D-branes probe the Planck scale beneath the perturbative string scale restores the former as the appropriate minimal resolvable length that ought to appear in any ‘generalized uncertainty principle’; although a proposal for a modified stringy uncertainty principle, valid for both strings and D-branes, can be found in [42].

  11. 11.

    One might regard the growth of the cross section for inelastic scattering (the effective target size for such scattering grows as k 1/2) as the weak coupling precursor of this behavior.

  12. 12.

    That 10d string theory is a limit of an 11d parent (dubbed M-theory) might be regarded as a variant of the mechanism that resolves singularities. The set of branes in the strongly coupled 11d theory are membranes and fivebranes; the limit of vanishing circle (1-cycle) size leads to light string-like excitations which are membranes wrapping the circle along one of its two worldvolume dimensions, with the other worldvolume coordinate parametrizing the motion of the string in the remaining nine spatial dimensions (with the string tension \(\mu_{\mathrm{str}}=\hbar c R_{\mathrm{circle}}/\ell_{p}^{3}\) becoming lighter and lighter as the circle size shrinks to well below the Planck scale in size).

  13. 13.

    In weak coupling, the masses of the m excitations are roughly given by the separation of the eigenvalues of the blocks x and y due to the commutator squared term in the Hamiltonian (4.11).

  14. 14.

    With the emergence of field theory as a realization of string theory, the subject has come full circle with a return to its roots—the gauge theory of QCD is the theory of the strong interactions, and gauge theory underlies string theory as well. The apparent mismatch of strong interaction effective dynamics with string theory turns out to be an artifact of expanding around the wrong vacuum [5860].

  15. 15.

    Also, under transformations such as T-duality, branes of different dimensions mix with one another, so the dimensionality of extended objects in spacetime has a certain degree of ambiguity.

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Martinec, E.J. Evolving Notions of Geometry in String Theory. Found Phys 43, 156–173 (2013). https://doi.org/10.1007/s10701-011-9609-5

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Keywords

  • String theory
  • Geometry
  • Topology
  • Quantum gravity